Math Is Fun Forum

A Newton of force accelerates 1 kg at 1 m/sec^2. The acceleration of gravity is 9.81 m/sec^2. So the force of gravity is 9.81 Newtons on 1kg of mass. Simply looking it up will confirm this.

Define down as vertical + and from P toward the pulley as horizontal +.
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Due to gravity pulling down on Q, it produces a horizontal force on P of +(3kg*9.81N/kg)=29.43N.   (1)

Due to gravity, P has a vertical force down on itself of +(2kg*9.81N/kg)=19.62N.   (2)

The 30N angled up force, produces
-(30N*cos(30°)) = -30N*0.866 = -25.98N horizontally   (3)
and -(30N*sin(30°))=-30N*0.5=-15N vertically.      (4)

So P has a horizontal force of
29.43N - 25.98N = 3.45N,   From (1)&(3) : (5)
and a vertical force of 19.62N - 15N = 4.62N. From (2)&(4) : (6)

The coefficient of friction is the horizontal force divided by the vertical force which is
3.45N/4.62N = 0.747 ≈ 0.75. From (5)&(6) : (7)
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You can treat P's and Q's mass together and determine the acceleration of P and Q with the total forces acting on them. Then the force needed to accelerate just P plus the friction would be the tension in the string.

Once the 30N force is removed, the force down on P is just due to gravity acting on its mass. Using the coefficient of friction this produces a horizontal force of
-(19.62N*0.747)=-14.65N. From(2)&(7) : (8)
The sign is negative since it resists being dragged toward the pulley.

Due to gravity, Q is producing a force of 29.43N. From (1).

The total force on P and Q together is
29.43N - 14.65N = 14.78N. From (1)&(8) : (9)
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F = ma so a = F/m = 14.78N/(3kg+2kg)
= (14.78kg-m/sec^2)/5kg = 2.956 m/sec^2. From(8) : (10)

Since P is 2kg and F = ma = 2kg * 2.956m/sec^2
= 5.912kg-m/sec^2 = 5.912N. From(10) : (11)

So with the friction on P and the force required to accelerate P, the tension in the string is
14.65N + 5.912N = 20.56N. From(8)&(11) : (12)
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We can double check by calculating the force due to gravity on Q that is left over based on it's acceleration.

If there were no tension in the string, in other words, nothing pulling up on Q it would accelerate down at 9.81m/sec^2. It is accelerating down at only 2.956m/sec^2, which is obviously the same as for P and Q together.

The resistance to being accelerated down is
3kg * 2.956m/sec^2 = 8.868N. From (10) : (13)

The tension in the string is the force of gravity on Q minus Q's resistance to being accelerated, which is
29.43N - 8.868N = 20.56N. From(1)&(13) : (14).

The double check is good, since both (12) and (14) produced a result of 20.56N.

bobbym,

Hmmm ..... except it sounded like the monthly interest rate was 0.5%, not 6%, while £100 was indeed being added monthly.

That is also showing compounding monthly. Compounding was not specified, but the explanation made it sound like it was not, which would be unusual, but is possible. Compounding is much more common.

The links gave a good explanation of compounding, but there is also a question as to whether the 6% is the effective annual, also just called annualized, rate, sometimes abbreviated APY (Annual Percentage Yield). If it is the APY, then the monthly rate, if it is compounded monthly, would be slightly lower than 0.5%. A monthly interest rate of 0.5% would give an apparent yield of more than 6% when compounded monthly over a year.

Hope I'm not complicating things too much. Compounding is much more usual, and using the effective annual rate isn't necesssary for understanding the basics of interest rates, but most investments and saving accounts list an effective rate, although some will list both.

"2-Hell VS Heaven (*)" could be hard, if a person just doesn't think exactly how to word the question. Once the right question is realized, then it seems too simple.

Correct? That was a neat puzzle. It might be as easy, or as hard, for a middle school student to answer it as for a much older adult to answer it. Either way, I still thought that it was a good puzzle.

Nice, and perhaps obviously, to solve for a and b,

If T(0)=0 and T(1)=1, then a=0 and b=0. However, if T(0) and T(1) are the initial conditions but are not given, then

If the course isn't this advanced, it might still be necessary to solve four simultaneous linear equations to solve for c, d, e, and f for n:even and n:odd, ignoring the a and b terms. The equations will be the same for n:even and n:odd if T(0)=0 and T(1)=1. If T(0) and T(1) are givens, but the values are not specified, T(0) and T(1) become terms in the solutions for n:even and n:odd, respectively, and the values of f are not the same for n:even and n:odd.

The solution to the puzzle presented in the first message in this thread appears to be.......

The proof is....

Note: Had to use Tom, Rick, Harry, and Sally as the names instead of the names given in the original Marble Mix Up puzzle.

Just like you, I did also. Our way is perhaps a more crude, but more direct, mathetmatical solution. It was the first thought that entered my mind and was the correct solution so it just stuck.

By the way, see
Click here to see message on hiding puzzle solutions in messages.
If you use the hide tag it will hide the solution that you just gave, just in case someone inadvertently sees the solution to the puzzle before they have tried to solve it for themselves. I did the same thing myself the first time I posted a message.

By the way, also, -ed is a common construction in English for the past tense, but "blowed" is not a word. The past tense of blow is "blew".